Infimum of Gamma distribution

Question

Let $X$ be a Gamma random variable with the CDF

$F_X(x)=\frac{1}{\Gamma(\alpha)}\gamma(\alpha,\beta x)$ where $\Gamma(x)$ represent the gamma function and $\gamma(a,b)$ denotes the lower-incomplete gamma function.

How to find the lower end, $v(F)$, of $F_X(x)$?, where $v(F)$ is

$v(F)=\inf\{x: F_X(x)>0\}$.

Answer 1

The Gamma distribution is supported on $[0,\infty)$, so $v(F) = 0$.